log-normal.md

Log-Normal Distribution (Base 10)

Distribution index · Natural-Space Ln-Normal · Parameterization crosswalk

Purpose and Parameterization

Numerics.Distributions.LogNormal models $Y=\log_{10}X$ as Normal and uses constructor order $(\mu_Y,\sigma_Y)$. Both parameters are expressed in base-10 logarithmic units; they are not the arithmetic mean and standard deviation of $X$. BestFit's standard instance uses Base = 10. Numerics requires a finite base greater than one and throws for invalid values. BestFit's standard convention and the equations below use base 10; changing the base requires converting both log-space parameters consistently.

Distribution

For $x>0$, write $y=\log_{10}x$, $z=(y-\mu_Y)/\sigma_Y$, and $c=\ln 10$. Then

$$ f_X(x)=\frac{\phi(z)}{x\sigma_Y c},\qquad F_X(x)=\Phi(z),\qquad S_X(x)=1-\Phi(z). \tag{LN10.1} $$

The factor $(x\ln10)^{-1}$ is the transformation Jacobian. The quantile is

$$ Q_X(p)=10^{\mu_Y+\sigma_Y\Phi^{-1}(p)}. \tag{LN10.2} $$

If $a=\sigma_Y\ln10$, then

$$ E[X]=\exp(\mu_Y\ln10+a^2/2),\quad \operatorname{Var}(X)=E[X]^2(e^{a^2}-1). \tag{LN10.3} $$

All positive moments exist. Median is $10^{\mu_Y}$; the upper tail is lognormal.

Likelihood and Inference

For exact positive values,

$$ \ell=-n\log\sigma_Y-n\log(\ln10)-\sum_i\log x_i -\frac n2\log(2\pi)-\frac12\sum_i z_i^2. \tag{LN10.4} $$

The Jacobian is essential for a likelihood on the original $X$ scale. RMC.BestFit's family methods supply the density and CDF to all exact, censored, uncertain, interval, threshold, stationary, and nonstationary components; there is no additional external transformation step. Trends act on $\mu_Y$ and $\sigma_Y$. The posterior multiplies the complete likelihood by parameter and quantile priors.

Initialization and working bounds are sample-dependent Numerics outputs and become default uniform-prior bounds. The log-space location $\mu_Y$ may be any finite real value; only the log-space standard deviation must be positive. Negative log means are therefore valid for positive observations below one in the stated measurement unit.

Compile-Checked API Example

private static (double Density, double Cdf, double Quantile) EvaluateLogNormal()
{
    var distribution = new global::Numerics.Distributions.LogNormal(3.20, 0.20);
    return Evaluate(distribution, value: 2200.0, nonExceedanceProbability: 0.99);
}

The mean 3.20 and standard deviation 0.20 describe $\log_{10}$ discharge. This is categorically different from the natural-space arguments in LnNormal.

Validation and Limitations

Closed-form verification uses a symmetric deterministic sample in base-10 log space. It verifies the population-divisor MLEs for $\mu_Y$ and $\sigma_Y$, the original-measure log likelihood including the $(x\ln 10)^{-1}$ Jacobian, equality of scalar and pointwise likelihood sums, the median CDF, and a selected analytical quantile. Parameter estimates agree within $10^{-5}$ and the maximized log likelihood within $10^{-8}$; direct likelihood identities use $10^{-10}$. Zero and negative observations have no support and must be represented through censoring or another family rather than silently transformed.

The distribution verification matrix identifies the current independent formula and fitted-objective comparisons, retained Bayesian/MLE recovery designs, and their distinct acceptance rules. The compiled example guards API compatibility; it does not run an estimator or establish scientific accuracy by itself.

References

[1] E. L. Crow and K. Shimizu, eds., Lognormal Distributions: Theory and Applications. Marcel Dekker, 1988.